#10166282

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#48647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,8)(3,5,10,15)(4,9,13,16)(6,14,12,11)(17,18)(19,27)(20,28)(21,25)(22,30)(23,24)(26,29)(31,32)(33,34)(35,38)(36,41)(37,46)(39,42)(40,45)(43,44)(47,48);(1,3)(2,5)(4,12)(6,13)(7,10)(8,15)(9,11)(14,16)(17,19,23,28)(18,20,24,27)(21,29,31,22)(25,30,32,26)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,48)(46,47);(2,8)(3,10)(4,13)(11,14)(18,24)(20,27)(25,32)(26,30)(33,36,39,46)(34,41,42,37)(35,43,45,47)(38,44,40,48);(1,4,7,13)(2,9,8,16)(3,12,10,6)(5,11,15,14)(17,20,23,27)(18,19,24,28)(21,30,31,26)(22,25,29,32)(33,37)(34,36)(35,44)(38,47)(39,41)(40,43)(42,46)(45,48);(1,5)(2,10)(3,8)(4,14)(6,16)(7,15)(9,12)(11,13)(17,21,23,31)(18,25,24,32)(19,22,28,29)(20,26,27,30)(33,34)(35,40)(36,41)(37,46)(38,45)(39,42)(43,48)(44,47);(1,6,7,12)(2,11,8,14)(3,13,10,4)(5,9,15,16)(17,22)(18,26)(19,21)(20,25)(23,29)(24,30)(27,32)(28,31)(33,38,39,40)(34,35,42,45)(36,44,46,48)(37,47,41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10073478);